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\centerline
{
\bf A Quick (.1 seconds!) Proof of Gigoujeu's  Two-Circle Theorem
}
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\centerline
{
\it Shalosh B. EKHAD
}
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\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
{\it \`A la m\'emoire de Michel J\'er\^ome Dufr\'enoy}
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In 1960, Francois N\'emorin Gigoujeu famously won the {\it grand prix} (a library of 3000 volumes!)
in the {\it concours} organized by the {\it Soci\'et\'e G\'en\'erale de Cr\'edit Instructionnel}
(see [1]), by solving the following  challenging {\it question}.

``{\it On donne deux circonf\'erences $OO'$: d'un point $A$ pris  sur $O$, on m\`ene des
tangentes \`a $O'$; on  joint les points de contact de ces tangentes: 
on m\`ene la tangente en $A$ \`a la circonf\'erence $O$; on demande le lieu du
point d'intersection de cette tangente avec la corde des contacts dans la circonf\'erence $O'$}''

By rescaling and rotating, we can let $O'$ be $x^2+y^2=1$ and $O$  be $(x-a)^2+y^2=r^2$. \hfill\break
Running the Maple code (by D. Zeilberger), available from 
{\eighttt http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/fng},
I immediately confirmed Gigoujeu's theorem that states that the desired locus is
$$
 {a}^{2} {x}^{4}+ 2a ( r^2- a^2-1 ) {x}^{3}+ 
\left ( {a}^{2}{y}^{2} + {a}^{4} + 4\,{a}^{2}  -2\,{a}^{2}{r}^{2} + (r^2 -1 )^2  \right ) {x}^{2}
+ 2a\,\left(  ( {r}^{2}- 1 ) {y}^{2}+{r}^{2}-{a}^{2}-1 \right) x
$$
$$
+ \left( (r^2 - 1)^2-{a}^{2}{r}^{2} \right) {y}^{2}+{a}^{2} \, = \, 0 \quad . \quad \halmos 
$$

See a diagram here: {\eighttt http://www.math.rutgers.edu/\~{}zeilberg/PG/Gigoujeu.html} .


{\bf Reference}

1. Jules Verne, ``{\it Paris au $XX^{e}$ si\`ecle}'', Hachette, 1994 .

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